I don't think either company is at risk of going out of business due to Amazon Air. I think what has changed is that Amazon Air isn't something that can easily be undersold and put out of business. So we will likely see three big shippers (I'm not counting DHL.) in the US.
I think any move to increase the effect of regulators in the US would be the biggest threat to any company, and Amazon is the most exposed. They're dominating enough diverse markets that a federally ordered breakup, like was done to Microsoft, is plausible. I don't see a stronger SEC in the near future regardless of which party holds the presidency or any house of congress, however.
Simply having a strong presence in a competitive shipping market actually does wonders to allay the concerns of regulators that would otherwise think about breaking up Amazon too.
I'm a pilot, and my degree was in natural science stuff. Someone with more perspective on government oversight and financial regulation can probably shoot all kinds of holes in my ideas.